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| Properties of the z-transform | ||||
| Time domain | Z-domain | Proof | ROC | |
| Notation | 
| 
| ROC: 
| |
| Linearity | 
| 
| 
| At least the intersection of ROC1 and ROC2 |
| Time expansion | 
 : integer
| 
| 
| R^{1/k} |
| Time shifting | 
| 
| 
| ROC, except  if  and  if 
|
| Scaling in the z-domain | 
| 
| 
| 
|
| Time reversal | 
| 
| 
| 
|
| Complex conjugation | 
| 
| 
| ROC |
| Real part | 
| 
| ROC | |
| Imaginary part | 
| 
| ROC | |
| Differentiation | 
| 
| 
| ROC |
| Convolution | 
| 
| 
| At least the intersection of ROC1 and ROC2 |
| Cross-correlation | 
| 
| At least the intersection of ROC of  and 
| |
| First difference | 
| 
| At least the intersection of ROC of X1(z) and 
| |
| Accumulation | 
| 
| 
| |
| Multiplication | 
| 
| - | |
| Parseval's relation | 
| 
|
, If 
causal
, Only if poles of 
are inside the unit circle
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| Examples conclusion | | | Relationship to Laplace transform |